The Entropy Shift and Extremality Shift at Zero Temperature

There is a well-established connection between the higher-derivative corrections to the black hole entropy and the black hole extremality bound. The particular combination of EFT coefficients $c_i$ that controls the mass shift at fixed charge and temperature also controls the entropy shift at fixed charge and mass in the limit where the mass approaches the uncorrected value for the extremal mass. In this note, we use the classical entropy function formalism to examine the entropy corrections at exactly zero temperature, or at the corrected value for the extremal mass. We find that the zero-temperature entropy shift (a) is unrelated to the mass shift, (b) is $\mathcal{O}(c_i)$ in the EFT coefficients, rather than $\mathcal{O}(\sqrt{c_i})$, as is the constant-mass entropy shift, and (c) is negative in the example of the EFT arising at low energies from QED plus gravity.